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Math 0300

Integers are the negative , zero, and positive numbers

An can be represented or graphed on a line by an arrow. An arrow pointing to the right represents a positive number. An arro w pointing to the left represents a negative num ber. The of the number can be determined by the distance to zero, or by counting the distance between numbers in the arrow. The integers 5 and –4 are shown on the in the figure below.

Example: +5 -4

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Example:

Another example of integers or positive and negative numbers is used with a checking account or dealing with money. A deposit to your checking account or pocket is an example of an addition called a positive integer (number or amount); a deduction or expense is an example of a negative integer (number/amount).

If there is a balance of \$25 in a checking account, and a check is written for \$30, the account will be overdrawn/overspent by \$5.

The sum of two integers can be shown on a number line. To add two integers, find the point on the number line corresponding to the first addend (integer). The sum is the number directly below the tip of the arrow.

Example: +2

4 + 2 = 6

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

-2

-4 +(-2) = - 6

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

+2

-4 + 2 = -2

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

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-2

4 + (-2) = 2

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

The of the addends can categorize the sums shown above.

The addends have the same .

4 + 2 = 6 positive 4 plus positive 2 equals 6 -4 + (-2)= -6 negative 4 plus negative 2 equals –6

-4 + 2 = -2 negative 4 plus positive 2 equals negative 2

4 + (-2) = -6 positive 4 plus negative 2 equals positive 2

The rule for adding two integers depends on whether the signs of the addends are the same or different.

Rule for Adding Tw o Integers:

TO ADD INTEGERS WITH THE SAME SIGN, add the absolute values of the numbers. Then attach the sign of the addends.

TO A DD IN TEGE RS W ITH D IFFER ENT S IGNS, find the ab solute values of the numbers. Subtract the smaller absolute value from the larger absolute value. Then attach the sign of the addend with the largest value.

Examples:

Add: (-4) +(-9) = -13 -14 + (-47) = - 61 6 + -13 = - 7 162 + (-247) = -85 -8 + 8 = 0

Note: in the last example we are adding a number and its opposite –8 + 8 = 0. The sum of a number and its is always zero.

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The properties of addition of whole numbers hold true for integers too.

The Addition Property of Zero a + 0 = a or 0 + a = a

The of Addition a + b = b + a

The of Addition (a + b) + = a + (b + c)

The Inverse Property of Addition a + (-a) = 0 or -a + a = 0

Add: (-4) + (-6) +(-8) + 9

Add the first two numbers (-4) + (-6) = (-10)

Add the sum to the third number (-10) + (-8) = (-18)

Continue until all the numbers have been added (-18) + 9 = -9

Example: Add the five changes in price of stock:

-2 + (-3) + (-1) + (-2) + (-1) = (-5) + (-1) +(-2) + (-1) = (-6) +(-2) + (-1) = -8 + (-1) = -9

The change is – 9. This means that the price of the stock fell \$9 per share.

Example:

Evaluate –x + y when x = -15 and y = -5. -x + y

Replace x with –15 and y with –5 -(-15) + (-5)

Simplify –(-15). =15 + (-5)

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Example:

Is – 7 a solution of the x + 4 = - 3?

Replace x by - 7 and then simplify. –7 + 4 = -3 -3 = -3

The results are equal. –7 is a solution of the equation.

Subtraction of integers

Recall that the sign – can indicate the sign of a number, as in –3 (negative 3), or can indicate the operation of subtraction, as in 9 – 3 (nine minus three).

Subtraction of integers can be written as the addition of the opposite number. To subtract two integers, rewrite the subtraction expression as the first number plus the opposite of the second number. Some examples are shown below.

First number - second number = First number + opposite of the second number

8 - 15 = 8 + (-15) = -7

8 - (-15) = 8 + 15 = 23

-8 - 15 = -8 + (-15) = -23

-8 - (-15) = -8 + 15 = 7

Rule for Subtracting Two Integers

To subtract two integers, add the opposite of the second integer to the first integer. This can be written symbolically as a - b = a + (-b).

Subtract (-15) – 75

Rewrite the subtraction operation as he sum of (-15) - 75 the first number and the opposite of the second = (-15) + (-75) number. The opposite of 75 is - 75. = - 90